Note on the Jordan form of an irreducible eventually nonnegative matrix

Date
2015-06-01
Authors
Hogben, Leslie
Tam, Bit-Shun
Wilson, Ulrica
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Mathematics
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Abstract

A square complex matrix A is eventually nonnegative if there exists a positive integer k(0) such that for all k >= k(0), A(k) >= 0; A is strongly eventually nonnegative if it is eventually nonnegative and has an irreducible nonnegative power. It is proved that a collection of elementary Jordan blocks is a Frobenius Jordan multiset with cyclic index r if and only if it is the multiset of elementary Jordan blocks of a strongly eventually nonnegative matrix with cyclic index r. A positive answer to an open question and a counterexample to a conjecture raised by Zaslavsky and Tam are given. It is also shown that for a square complex matrix A with index at most one, A is irreducible and eventually nonnegative if and only if A is strongly eventually nonnegative.

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This article is published as Hogben, Leslie, Bit-Shun Tam, and Ulrica Wilson. "Note on the Jordan form of an irreducible eventually nonnegative matrix." The Electronic Journal of Linear Algebra 30 (2015): 279-285. DOI: 10.13001/1081-3810.3049. Posted with permission.

Keywords
Irreducible eventually nonnegative, Strongly eventually nonnegative, Eventually reducible, Eventually r-cyclic, Cyclic index, Frobenius collection, Frobenius Jordan multiset, Jordan multiset, Jordan form
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